To solve the system of equations
[tex]\[ y = x^2 + 2x + 7 \][/tex]
[tex]\[ y = x + 7 \][/tex]
let's set the two equations equal to each other since they both equal [tex]\( y \)[/tex]:
[tex]\[ x^2 + 2x + 7 = x + 7 \][/tex]
Subtract [tex]\( x + 7 \)[/tex] from both sides to set the equation to zero:
[tex]\[ x^2 + 2x + 7 - (x + 7) = 0 \][/tex]
[tex]\[ x^2 + 2x + 7 - x - 7 = 0 \][/tex]
[tex]\[ x^2 + x = 0 \][/tex]
Factor out [tex]\( x \)[/tex]:
[tex]\[ x(x + 1) = 0 \][/tex]
This gives us two solutions for [tex]\( x \)[/tex]:
[tex]\[ x = 0 \][/tex]
[tex]\[ x = -1 \][/tex]
Next, we need to find the corresponding [tex]\( y \)[/tex] values for these [tex]\( x \)[/tex] solutions by substituting back into the simpler original equation [tex]\( y = x + 7 \)[/tex]:
For [tex]\( x = 0 \)[/tex]:
[tex]\[ y = 0 + 7 \][/tex]
[tex]\[ y = 7 \][/tex]
So one solution is [tex]\( (0, 7) \)[/tex].
For [tex]\( x = -1 \)[/tex]:
[tex]\[ y = -1 + 7 \][/tex]
[tex]\[ y = 6 \][/tex]
So the other solution is [tex]\( (-1, 6) \)[/tex].
Therefore, the solution set is [tex]\((0, 7)\)[/tex] and [tex]\((-1, 6)\)[/tex].
Among the given choices, the correct one is:
[tex]\( \boxed{D. (0, 7) \text{ and } (-1, 6)} \)[/tex]
